Library UniMath.CategoryTheory.limits.terminal
- A proof that the terminal object is a property in a (saturated/univalent) category (isaprop_Terminal)
- Construction of the terminal object from the empty product (terminal_from_empty_product)
Require Import UniMath.Foundations.PartD.
Require Import UniMath.Foundations.Propositions.
Require Import UniMath.Foundations.Sets.
Require Import UniMath.MoreFoundations.Tactics.
Require Import UniMath.CategoryTheory.total2_paths.
Require Import UniMath.CategoryTheory.Categories.
Require Import UniMath.CategoryTheory.functor_categories.
Require Import UniMath.CategoryTheory.limits.products.
Require Import UniMath.CategoryTheory.Monics.
Local Open Scope cat.
Section def_terminal.
Context {C : precategory}.
Definition isTerminal (b : C) : UU := ∏ a : C, iscontr (a --> b).
Definition Terminal : UU := ∑ a, isTerminal a.
Definition TerminalObject (T : Terminal) : C := pr1 T.
Coercion TerminalObject : Terminal >-> ob.
Definition TerminalArrow (T : Terminal) (b : C) : b --> T := pr1 (pr2 T b).
Lemma TerminalArrowUnique {T : Terminal} {a : C} (f : C⟦a,TerminalObject T⟧) :
f = TerminalArrow T _.
Proof.
exact (pr2 (pr2 T _ ) _ ).
Defined.
Lemma TerminalEndo_is_identity {T : Terminal} (f : T --> T) : f = identity T.
Proof.
now apply proofirrelevance, isapropifcontr, (pr2 T T).
Qed.
Lemma TerminalArrowEq {T : Terminal} {a : C} (f g : a --> T) : f = g.
Proof.
now rewrite (TerminalArrowUnique f), (TerminalArrowUnique g).
Qed.
Definition mk_Terminal (b : C) (H : isTerminal b) : Terminal.
Proof.
∃ b; exact H.
Defined.
Definition mk_isTerminal (b : C) (H : ∏ (a : C), iscontr (a --> b)) :
isTerminal b.
Proof.
exact H.
Defined.
Lemma isiso_from_Terminal_to_Terminal (T T' : Terminal) :
is_iso (TerminalArrow T T').
Proof.
apply (is_iso_qinv _ (TerminalArrow T' T)).
now split; apply TerminalEndo_is_identity.
Defined.
Definition iso_Terminals (T T' : Terminal) : iso T T' :=
(TerminalArrow T' T,,isiso_from_Terminal_to_Terminal T' T) .
Definition hasTerminal := ishinh Terminal.
Section Terminal_Unique.
Hypothesis H : is_univalent C.
Lemma isaprop_Terminal : isaprop Terminal.
Proof.
apply invproofirrelevance.
intros T T'.
apply (total2_paths_f (isotoid _ H (iso_Terminals T T')) ).
apply proofirrelevance.
unfold isTerminal.
apply impred.
intro t ; apply isapropiscontr.
Qed.
End Terminal_Unique.
End def_terminal.
Arguments Terminal : clear implicits.
Arguments isTerminal : clear implicits.
Arguments TerminalObject {_} _.
Arguments TerminalArrow {_} _ _.
Arguments TerminalArrowUnique {_} _ _ _.
Arguments mk_isTerminal {_} _ _ _.
Arguments mk_Terminal {_} _ _.
Section Terminal_and_EmptyProd.
Require Import UniMath.Foundations.Propositions.
Require Import UniMath.Foundations.Sets.
Require Import UniMath.MoreFoundations.Tactics.
Require Import UniMath.CategoryTheory.total2_paths.
Require Import UniMath.CategoryTheory.Categories.
Require Import UniMath.CategoryTheory.functor_categories.
Require Import UniMath.CategoryTheory.limits.products.
Require Import UniMath.CategoryTheory.Monics.
Local Open Scope cat.
Section def_terminal.
Context {C : precategory}.
Definition isTerminal (b : C) : UU := ∏ a : C, iscontr (a --> b).
Definition Terminal : UU := ∑ a, isTerminal a.
Definition TerminalObject (T : Terminal) : C := pr1 T.
Coercion TerminalObject : Terminal >-> ob.
Definition TerminalArrow (T : Terminal) (b : C) : b --> T := pr1 (pr2 T b).
Lemma TerminalArrowUnique {T : Terminal} {a : C} (f : C⟦a,TerminalObject T⟧) :
f = TerminalArrow T _.
Proof.
exact (pr2 (pr2 T _ ) _ ).
Defined.
Lemma TerminalEndo_is_identity {T : Terminal} (f : T --> T) : f = identity T.
Proof.
now apply proofirrelevance, isapropifcontr, (pr2 T T).
Qed.
Lemma TerminalArrowEq {T : Terminal} {a : C} (f g : a --> T) : f = g.
Proof.
now rewrite (TerminalArrowUnique f), (TerminalArrowUnique g).
Qed.
Definition mk_Terminal (b : C) (H : isTerminal b) : Terminal.
Proof.
∃ b; exact H.
Defined.
Definition mk_isTerminal (b : C) (H : ∏ (a : C), iscontr (a --> b)) :
isTerminal b.
Proof.
exact H.
Defined.
Lemma isiso_from_Terminal_to_Terminal (T T' : Terminal) :
is_iso (TerminalArrow T T').
Proof.
apply (is_iso_qinv _ (TerminalArrow T' T)).
now split; apply TerminalEndo_is_identity.
Defined.
Definition iso_Terminals (T T' : Terminal) : iso T T' :=
(TerminalArrow T' T,,isiso_from_Terminal_to_Terminal T' T) .
Definition hasTerminal := ishinh Terminal.
Section Terminal_Unique.
Hypothesis H : is_univalent C.
Lemma isaprop_Terminal : isaprop Terminal.
Proof.
apply invproofirrelevance.
intros T T'.
apply (total2_paths_f (isotoid _ H (iso_Terminals T T')) ).
apply proofirrelevance.
unfold isTerminal.
apply impred.
intro t ; apply isapropiscontr.
Qed.
End Terminal_Unique.
End def_terminal.
Arguments Terminal : clear implicits.
Arguments isTerminal : clear implicits.
Arguments TerminalObject {_} _.
Arguments TerminalArrow {_} _ _.
Arguments TerminalArrowUnique {_} _ _ _.
Arguments mk_isTerminal {_} _ _ _.
Arguments mk_Terminal {_} _ _.
Section Terminal_and_EmptyProd.
Construct Terminal from empty arbitrary product.
Definition terminal_from_empty_product (C : precategory) :
Product empty C fromempty → Terminal C.
Proof.
intros X.
use (mk_Terminal (ProductObject _ C X)).
use mk_isTerminal.
intros a.
assert (H : ∏ i : empty, C⟦a, fromempty i⟧) by
(intros i; apply (fromempty i)).
apply (iscontrpair (ProductArrow _ _ X H)); intros t.
apply ProductArrowUnique; intros i; apply (fromempty i).
Defined.
End Terminal_and_EmptyProd.
Product empty C fromempty → Terminal C.
Proof.
intros X.
use (mk_Terminal (ProductObject _ C X)).
use mk_isTerminal.
intros a.
assert (H : ∏ i : empty, C⟦a, fromempty i⟧) by
(intros i; apply (fromempty i)).
apply (iscontrpair (ProductArrow _ _ X H)); intros t.
apply ProductArrowUnique; intros i; apply (fromempty i).
Defined.
End Terminal_and_EmptyProd.
Section TerminalFunctorCat.
Variables (C D : precategory) (ID : Terminal D) (hsD : has_homsets D).
Definition Terminal_functor_precat : Terminal [C,D,hsD].
Proof.
use mk_Terminal.
- use mk_functor.
+ use tpair.
× intros c; apply (TerminalObject ID).
× simpl; intros a b f; apply (TerminalArrow ID).
+ split.
× intro a; apply TerminalEndo_is_identity.
× intros a b c f g; apply pathsinv0, TerminalArrowUnique.
- intros F.
use tpair.
+ use mk_nat_trans; simpl.
× intro a; apply TerminalArrow.
× intros a b f; simpl.
rewrite (TerminalEndo_is_identity (TerminalArrow ID ID)), id_right.
apply TerminalArrowUnique.
+ abstract (intros α; apply (nat_trans_eq hsD); intro a; apply TerminalArrowUnique).
Defined.
End TerminalFunctorCat.
Variables (C D : precategory) (ID : Terminal D) (hsD : has_homsets D).
Definition Terminal_functor_precat : Terminal [C,D,hsD].
Proof.
use mk_Terminal.
- use mk_functor.
+ use tpair.
× intros c; apply (TerminalObject ID).
× simpl; intros a b f; apply (TerminalArrow ID).
+ split.
× intro a; apply TerminalEndo_is_identity.
× intros a b c f g; apply pathsinv0, TerminalArrowUnique.
- intros F.
use tpair.
+ use mk_nat_trans; simpl.
× intro a; apply TerminalArrow.
× intros a b f; simpl.
rewrite (TerminalEndo_is_identity (TerminalArrow ID ID)), id_right.
apply TerminalArrowUnique.
+ abstract (intros α; apply (nat_trans_eq hsD); intro a; apply TerminalArrowUnique).
Defined.
End TerminalFunctorCat.
Morphisms from the terminal object are monic
Section monics_terminal.
Context {C : precategory} (TC : Terminal C).
Lemma from_terminal_isMonic (a : C) (f : C⟦TC,a⟧) : isMonic f.
Proof.
apply mk_isMonic; intros b g h H.
now apply TerminalArrowEq.
Qed.
End monics_terminal.
Context {C : precategory} (TC : Terminal C).
Lemma from_terminal_isMonic (a : C) (f : C⟦TC,a⟧) : isMonic f.
Proof.
apply mk_isMonic; intros b g h H.
now apply TerminalArrowEq.
Qed.
End monics_terminal.